Question

Consider the function f(x) = 12x⁵ + 60x⁴ - 100x³ + 2. For this function, there are four important intervals: (-[infinity], A], [A, B], [B, C], and [C, [infinity]) where A, B, and C are the critical numbers. Find A.

Answer 1

To find the critical numbers of the function f(x) = 12x⁵ + 60x⁴ - 100x³ + 2, we need to find the values of x where the derivative of f(x) is equal to zero or undefined. By finding the derivative of f(x) and setting it equal to zero, we can solve for the critical number A. The critical numbers of the function are A = 0, B ≈ -2.638, and C ≈ 1.971.

Step-by-step explanation:

To find the critical numbers of the function f(x) = 12x⁵ + 60x⁴ - 100x³ + 2, we need to find the values of x where the derivative of f(x) is equal to zero or undefined. The critical numbers occur at the values of x where f'(x) = 0 or f'(x) is undefined. Therefore, we need to find the derivative of f(x) and set it equal to zero to solve for A.

1. First, let's find the derivative of f(x). Taking the derivative of each term separately, we get f'(x) = 60x⁴ + 240x³ - 300x².
2. Next, set f'(x) equal to zero and solve for x: 60x⁴ + 240x³ - 300x² = 0.
3. Now we can factor out x² to simplify the equation: x²(60x² + 240x - 300) = 0.
4. Using the zero-product property, we know that either x² = 0 or 60x² + 240x - 300 = 0.
5. Solving x² = 0, we find that x = 0 is a critical number.
6. Solving 60x² + 240x - 300 = 0, we can use the quadratic formula to find the other critical numbers. The quadratic formula gives us x = (-b±√(b²-4ac))/(2a), where a = 60, b = 240, and c = -300.
7. After solving the quadratic equation, we find two possible values for x: x ≈ -2.638 and x ≈ 1.971.

Therefore, the critical numbers of the function f(x) = 12x⁵ + 60x⁴ - 100x³ + 2 are A = 0, B ≈ -2.638, and C ≈ 1.971.